![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > funcnv0 | Structured version Visualization version GIF version |
Description: The converse of the empty set is a function. (Contributed by AV, 7-Jan-2021.) |
Ref | Expression |
---|---|
funcnv0 | ⊢ Fun ◡∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fun0 6616 | . 2 ⊢ Fun ∅ | |
2 | cnv0 6144 | . . 3 ⊢ ◡∅ = ∅ | |
3 | 2 | funeqi 6572 | . 2 ⊢ (Fun ◡∅ ↔ Fun ∅) |
4 | 1, 3 | mpbir 230 | 1 ⊢ Fun ◡∅ |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4322 ◡ccnv 5673 Fun wfun 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-mo 2529 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-fun 6548 |
This theorem is referenced by: f10 6868 pthdlem1 29700 0trl 30052 0pth 30055 |
Copyright terms: Public domain | W3C validator |